Structural equivalence focuses our attention on pair-wise comparisons of actors. By trying to find actors who can be swapped for each other, we are really paying attention to the positions of the actors in a particular network. We are trying to find actors who are clones or substitutes.
Automorphic equivalence begins to change the focus of our attention, moving us away from concern with individual's network positions, and toward a more abstracted view of the network. Automorphic equivalence asks if the whole network can be re-arranged, putting different actors at different nodes, but leaving the relational structure or skeleton of the network intact.
Suppose that we had 10 workers in the University Avenue McDonald's restaurant, who report to one manager. The manager, in turn, reports to a franchise owner. The franchise owner also controls the Park Street McDonald's restaurant. It too has a manager and 10 workers. Now, if the owner decided to transfer the manager from University Avenue to the Park Street restaurant (and vice versa), the network has been disrupted. But if the owner transfers both the managers and the workers to the other restaurant, all of the network relations remain intact. Transferring both the workers and the managers is a permutation of the graph that leaves all of the distances among the pairs of actors exactly as it was before the transfer. In a sense, the "staff" of one restaurant is equivalent to the staff of the other, though the individual persons are not substitutable.
The hypothetical example of the restaurants suggests the main utility of the automorphic equivalence concept. Rather than asking what individuals might be exchanged without modifying the social relations described by a graph (structural equivalence), the somewhat more relaxed concept of automorphic equivalence focuses our attention on sets of actors who are substitutable as sub-graphs, in relation to other sub-graphs. In many social structures, there may well be sub-structures that are equivalent to one another (or approximately so). The number, type, and relations among such sub-structures might be quite interesting. Many structures that look very large and complex may actually be composed (at least partially) of multiple identical sub-structures; these sub-structures may be "substitutable" for one another. Indeed, a McDonalds is a McDonalds is a McDonalds...